where \(\tau \)  is the integration time. Simplified expressions for its evaluation at each of the frequency ranges defined above can be found in [1]}. The phase noise at each frequency range can be converted to Allan deviation for a corresponding integration time range. The phase noise for high frequencies (\(f\gg \chi f_c\) ) determines the Allan deviation for short integration times below certain value \(\tau _c /\chi \) . Therefore, by introducing () in (REF ), we obtain
\(\sigma _y(\tau \ll \tau _c/\chi )=\frac{1}{a_x} \sqrt{\frac{k_B T}{ m Q \omega _0^3 \tau }} = \sigma _{y0}.\) 
